Comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices
Let \(K\) be a proper cone in \(\mathbb{R}^n\) and call the real \(n\times n\) matrix \(A\) \(K\)-nonnegative if \(AK\subseteq K\) writing \(A\geq 0\) in this case. A splitting \(A= M- N\) (\(M\) nonsingular) is called weak nonnegative if \(M^{-1}\geq 0\) an \(M^{-1}N\geq 0\) or if \(M^{-1}\geq 0\) and \(NM^{-1}\geq 0\). Denoting by \(\rho(A)\) the spectral radius of \(A\) the authors present conditions on two weak nonnegative splittings \(A= M_1- N_1= M_2- N_2\) such that the comparison result \[ \rho(M^{-1}_1 N_1)< \rho(M^{-1}_2 N_2)< 1\tag{1} \] holds with the first `\(<\)' being replaced by `\(\leq\)' in a part of the cases. As a main result they show that some of their conditions can be retrieved if (1) holds together with appropriate additional assumptions.
scientific article; zbMATH DE number 1419875